Entrelacs abound in knitting at the moment. In thinking about perhaps composing my thoughts for a post on the subject, I searched online and found some very good sources on this subject. One is found at howtoknitasweater.com. The author, Cheryl Brunette also shares an article on lace . Here are 2 of my teaching samples, executed on 4.5 mm machine
front view
a rearview with ends woven in
…
Removing a circle from the center of our pie yields the “donut”. The purpose and size of the “donut hole” can vary from the size needed to apply a central motif whether in knit, crochet, or other forms, to one large enough to allow for insertion of a “back piece” that can be anything from a “square” to one that included a bit of shoulder, armhole, even neck shaping and an optional curve at the waist/hip area. 
Additional rows of plain knitting in the “donut” itself alter the final forms. Seaming can occur where preferred; the direction of pattern repeats if in use further influence choice of seam placement.
If miter shapes are created in the knitting method, the corners of the triangles will want to “poke out”. This can be a purposeful design feature. If they are not wanted one way to soften them is to have stitches for at least an inch at the outer circumference of the circle knitting with no shaping in that area, adding a border, going the spiral route. Swatching helps determine preference in creating personal designs. Small scale paper collages sorting out geometric shapes and joins can inspire the large form variations.
Vests and sweaters built on circular shapes offer some challenges. Shawls and shoulder wraps are much more forgiving, but garments, particularly if sleeves are added, can provide sizing and fitting challenges. Making a muslin in disposable knit yardage prior to the actual knitting allows for trial placement of armholes and testing of overall measurements prior to charting out garment and sleeve shaping. Slits for the armholes can easily be taped or stitched closed to suit, and in turn, trial cut in a different location. Trimming or adding borders to circumference allows for visualizing size grading. This process helps spare the knitter regrets upon completion of the piece.
Drawing large circles is easy and accurate with a “yardstick compass”. Trammel points are available online and at many woodworking supply stores, etc. They convert any standard size yardstick for drawing arcs and circles up to 72 inches dia. Use a longer stick the same width and thickness as a yardstick and draw circles as large as you like. They are usually made of aluminum except for the steel point, measure about 3-1/2 inches in length.
Some beginning guidelines for drafting:
1. Use your bust measurement as the circle’s diameter and draw the corresponding shape. Two or more strips of freezer paper may be used as the drawing surface, temporarily fused onto the knit yardage, becoming the paper “pattern” for the piece and stabilizing the knit for cutting.
2. Measure your back from arm to arm to determine how far apart to place armholes or obtain this measurement from any well-fitting favorite.
3. Measure armhole depth from the top of the shoulder to 2-3 inches below the armpit.
4. Draw lines for armholes and center horizontally within the body of the circle, shoulder measurement apart. Commercially written patterns are bountiful online and in magazines and tend to center the armholes vertically as well. I prefer them shifted up for a less bulky “collar”, and for placement of sleeves with raglan or traditional caps. Binding off a few stitches at the base and casting them on at the top of the slit create a slightly shaped for easing in the sleeve top.
5. Cut “armholes”, remove freezer paper if used, try on for fit, adjust as needed. 6. Back to more math!
I made a series of long-sleeved circular sweaters for sale in 2008. Discovered problems with photos in my photo library (new computer). These are an attempt at “restored” shots of one of the first such sweaters. The yarn was fine Italian mohair.
close up
Holding/ short-rowing is used to knit each wedge. The basic rule for holding is followed: stitches are brought in to work on the carriage side, and into hold opposite the carriage to avoid floats. The greater the number of sections, for either the full circle or a donut, the smoother the curves at the outer circumference, as can be imagined in the form below, which divided into more sections than those 5 in our original calculation.
The pie slices” begin and end on open stitches, ultimately requiring a join where the radii meet, using whatever method is preferred by the knitter. Assuming the knit carriage is on left and set for hold in the diagrams below if all stitches are brought into hold except 1 on the carriage side (or number required by calculations for the individual piece), 2 rows are knit, and action is repeated until all needles are in work, the following shape starts to fill in and a miter with pointed edges is created.
If all the needles are in work, and one begins to bring them into hold opposite the carriage, one begins to fill in the shape creating the form below, and a spiral is created, with circumference edges more rounded.
Like shapes may be stacked sequentially. If shaping at the top of the initial triangle wedge is reversed, however, the following begins to occur.
Knit rows in between the triangles begin to create larger holes in the center of the pie, and a donut occurs. The donut center can be varied, the knit rows between triangles increased to suit. The illustration below shows some of the variables. The plain knit rows are another factor in smoothing that outer circumference.
There are many knitting programs that will perform the necessary calculations, as well as a variety of knit calculators. The diophantine formula is the basis for what is known to some knitters as the “magic formula”. In the early 1980s, Alles Hutchinson authored a small book on the subject. There is a bit of personal leeway in the results, and the formula may be used in calculating even complex shapes with the proviso that one has the patience to break such shapes into series of simpler ones.
There are many online resources for information and calculators to sort out the math, including a triangle calculator. The original website’s offerings are now closed, but the info remains available here
https://web.archive.org/web/20200224005535/http://www.getknitting.com/ak_0603triangle.aspx
Using the gauge to match the previous post of 4S and 6R per inch the calculation for the pie divided into five triangles breaks down into the web calculator result pictured below:
The longhand method for the same calculation follows and also translates to: bring into hold 2 stitches for 4 times, 1 stitch for 80 times. Stitches in shaping are proofed as above: 88 stitches shaped over 84 rows.
Math is not always fun and is downright dreaded by some. One instance in knitting wherein basic calculations are required is in obtaining stitch and row gauges. I have known one hand knitter who would purchase yarn (not necessarily the one used in pattern), knit happily away, and try the finished product on everyone she knew until she found an accommodating body shape and size. If a large number of family and friends did not oblige, sweaters were stored until such a correct body appeared. Predictable results require careful measurements and some basic formula calculations.
Using home knitting machines to produce circular forms one resorts to breaking down the round object into pie wedges, which in turn are knit as triangles with straight line outer edges. The outer final circumference curve is controlled in a number of ways, one is by creating a far greater number of pie slices. For this exercise, I will work with 5 segments. 
There are some math constants. One example: to find the circumference of a circle its diameter is multiplied by pi = 3.14. If the diameter of our knit is 44 inches, its circumference will measure 44 X 3.14 = 138.16 inches. Using the rule of 5 or less than 5, this measurement is rounded to 138 inches.
The radius becomes the width of the pie wedge. In this instance, it would measure 22 inches. Let us assume our gauge is 4 stitches and 6 rows per inch. The radius is converted to stitches: 22 X 4 = 88 sts. The circumference becomes rows: 138 X 6 = 828 rs. If subdivided into 5 slices, each slice would be composed of 166 rows.
To knit the pie slice, short rows are used; since they happen every 2 rows, our row number for outer edge is divided by 2, yielding the total of now 83, which in this exercise I will round off to the even # 84.
Circular sweaters and vests have been in the pattern marketplace for a while, and there are very few online resources for purchasing patterns for machine knitting. My first attempt at one such pattern was a hand-knit for my granddaughter. It began on long circulars and was worked from the outer diameter, with stitches decreased at intervals, eventually bound off at the center. Most adult sweaters and vests both in HK and crochet are worked from the center out. A common diagram for such sweaters regardless of approach is seen below, a vest could result by simply omitting the sleeves.
Diagram A
Placement and shapes of sleeves are crucial to fit. A straight sleeve top as seen in dropped shoulder sweaters can result in the sleeve opening occurring on the arm inches below the shoulder, making the top of the circle that forms the collar flop around, and the sweater is hard to keep on (the result in my hand knit). Better shoulder fit and having a cap at the top of the sleeve, whether traditional or raglan and adjusting back garment width can achieve stability. In large sizes created by simply making the circle larger, the fit becomes very different in the front, and there are other, better ways to achieve a similar look. Calculations involved in planning these garments share common methods with “doilies”, shawls, ruffles, and more.
In hand knit/crochet another approach is to knit a square/rectangle variant, which can begin and end in open stitches. This piece is folded in half, circulars are used to pick up all the open stitches which now become the inner diameter of a circle, and increases are made regularly, evenly across the rows at intervals to achieve the desired outer diameter. The opening created by the folded fabric becomes the armhole, which may have to be partially stitched closed upon completion of the piece depending on size requirements.
Diagram B: red line represents fold
On the home knitting machines evenly spaced increases and decreases are possible but not practical, requiring removal of the kniting whether on garter bar or ravel cord, with rehanging of all the stitches after adjusting the number of needles in use. Patterning can continue if this is done on plain knit rows, and design shifts are taken into consideration. Though circular knits can be achieved using a ribber, there are distinct differences in tension between the two beds in Japanese machines, not so in Passap knitting. Here again, increasing/decreasing evenly across rows impractical if not impossible. Increases along the outer edges of the knit are easy, but the result is a series of triangles meeting and pointing down from the garment on that edge, a common sight in the marketplace at the moment, but not so good if a circle is the desired shape. One approach on the machine can be seen in the diagram below. The central shape is shared with the above diagram, but the circle is broken up so the garment becomes a flat construct. “Extra” smaller rectangles on the side represent a possible longer cap sleeve. Here the whole piece would be folded in half and seamed. If the center shape has straight sides adjustments can be made in seaming for the desired armhole measurement. The remaining problem: how to get that outer “diameter” created by the bottom and top of the piece to approach the width needed without increases and decreases across rows. With this approach the seams that bring the top and bottom sections together will move toward the front of the garment, so the joining method and its visibility is a consideration.
Diagram C
There are a number of ways to “play with” this technique, transfer lace on these machines becomes a hand technique aided by needle selection made by the card reader, based on punched holes mylar markings.
Method 1: single carriage using published cards that are for lace carriages that transfer only (Brother, Knitking, Toyota, Studio/Knitmaster fashion lace started on the appropriate row ).
1. Lock card on the first row as would be done on a standard machine.
2. KH carriage starts out on the Left side with end needle selection canceled.
3. Remove yarn from the feeder, store on left. Deactivate the row counter (flip automatic tripper back ).
4. First selection row is made left to right. Transfers can then be made by hand in direction of arrows, moving stitches on the selected needles to their left, away from the carriage. All needles must be returned to the B position.
5. Push in both part (slip) buttons.
One thing to keep in mind: when the KH carriage is set to slip, it is the selected needles that actually knit stitches. If the carriage moves across those same needles with no yarn in the feeder, selected needles will drop their corresponding stitches.
6. Set card to advance “normally”, move the carriage to the left.
7. Continue to make transfers that the lace carriage would have made in the direction as marked on the card until knit row markings are reached and no needles are selected.
8. Release part buttons, leave KC engaged.
9. Put yarn back in the feeder and the row count tripper back into work or advance RC by hand for the required number of knit rows
10. Knit 2 rows or if lace pattern requires 4 or more rows of knitting between repeats KC needs to be turned off for 2 rows less than needed, then turned back on for the last 2 knit rows. A lot to keep track of, but still quicker than hand knitting. Forgetting to put yarn in the feeder for knit rows or to set carriage back to slip for transfer selection rows will result in what I have come to fondly refer to as drop-itis.
Method 2: using 2 carriages, extension rails.
Bulky extension rails are shaped differently than those for the standard KM, their front “arm” quite a bit longer than the rear one. Rails are handy when knitting very wide pieces, a necessity when multiple carriages are in use, and especially when both are set to KC and engage the belt.
This method’s movements mimic those of traditional lace knitting in the standard Brother machine.
1. The “lace” carriage on left is set to slip in both directions, its RC tripper is in the off position.
2. The KH carriage on the right set to knit, its row count tripper on.
3. End needle selection is canceled on both carriages.
4. Left carriage operates “like” the lace carriage in terms of needle selection, but it is the knitter who hand manipulates the transfers, which are made in the same direction as the marked arrows on the card, away from the carriage.
5. When no needles are selected and the black arrow markings for knit rows on the lace card is reached, the “Lace” carriage is returned to its position off the bed on the left
6. The R, KH carriage then knits required rows, returns to rest on the R rail, and the process begins over again.
Since the KH carriage is not on KC and the card does not advance on knit rows, no adjustments need to be made to its settings if multiple knit rows are required.
Method 3: using studio simple lace cards (those cards that are not “suitable” for brother KMs). These cards are intended for machines that select and knit with each carriage pass
1. Using Studio card with A-side up lock card on row 3.
2. Knit carriage: KC in use, cancel end needle selection, no cam buttons throughout (this results in needle selection, but fabric created is in stocking stitch).
3. Row counter on 3
4. Make the first selection row right to left. Transfer stitches selected onto adjacent needles toward the same side as carriage (in this instance from right to left). All needles must be returned to the B position.
5. Release card, set card to advance normally. After each selection row, hand transfers continue to be made toward the carriage.
I found it easier to track the direction of transfers by remembering to move toward the carriage than to follow the direction of arrow markings on the card which do match, with even-numbered rows transferred toward the right, odd-numbered ones toward left.
The simple lace cards may also be used on the standard punchcard brother KM. Two carriages are selecting needles, so usual precautions for such knitting must be used, with each carriage off the bed and on the extension rail while the alternate carriage moves toward it from the opposite side. As the carriages are switched, operating from the opposite side, the previous selection is repeated.
1. Studio punchcard is locked on Row 3, KC (knit carriage) on right. The main carriage is set to KC, end needle selection is canceled, no cam buttons are used, it will knit with every pass it makes and selects the needles for the subsequent transfer row.
2. Release card, knit one row from right to left
3. KCOL, the Lace Carriage is engaged on the opposite Right side and moves toward the knit carriage to make the transfers. As it travels across the bed it will reselect the same needles (now empty). After transfers are completed, make a second pass with the lace carriage, and remove it from the needle bed.
5. The KH carriage is now free to make its pass for the single row of knitting to the opposite side, from Left to Right.
6. The lace carriage is engaged opposite to the knit carriage (now the left) and makes transfers toward it once again. It makes a second pass to its original place on right and is removed from the bed to be used again from the left. The process is repeated for the pattern. The Studio simple lace card may of course be redrawn with extra rows needed for standard use on the Brother KM.
the punchcard 
the resulting transfers on both the bulky and standard Brother machines
Extension rails for the bulky machine are available for purchase, are handy when producing wider pieces and particularly if knitting on the whole bed. Be aware if purchasing them, that the rails for the 9 mm machine are a different size, noticeable in the photo below. The bulky rail has a white tag on it
The following illustrate some of the processes involved in planning out fabric akin to the one in the previous post. Black borders outline blocks of 6 stitches/rows, reflect markings on blank Brother cards. In option A: the motif is planned and drawn. In this instance, it is colored in green (1), the area it covers will ultimately remain unpunched on the card or blank on mylar. Graph paper may be used to work this out, knit design software, or as in this case, an excel spreadsheet. A grid is created with every other square blacked out or colored in (2). The motif is superimposed on grid (3). Repeat is expanded adding 2 blank rows above each design row (4). Rust squares represent punched holes in the card or black squares in mylar.
Option A
Option B: the same motif is lengthened X3 (5). The lace mesh base is drawn out (6). The elongated motif is then superimposed on mesh (7). Electronic patterning on 910 allows for minimal drawing using all black squares, in turn making it necessary to color reverse for lace. Two separate motifs are used, the method for programming such repeats is in 910 manual. However, there are considerations for needle position and pattern selector placement for this “shortcut” to work properly, the steps are described by Kathleen Kinder and others. I prefer to work with what I “see” in terms of punched holes or squares, this is the method used to develop my previous flower motif swatch. The short supply of mylars may also be a consideration in using them or not for such large, and perhaps limited use design repeats. If interface cables and software are available, other options abound.
Option B
Many articles were written in the 1980s in Australia, New Zealand, and Britain, some finding their way to newsletters published in the USA at the time investigating this subject. With the advent of electronics the process became “easier”. Kathleen Kinder author of several books on Machine Knitting covering myriad topics (one whole text on lace), also authored Electonic Knitting: an Introduction for Brother and Knitmaster Knitters (?1989) that investigated the move from punchcard to electronics, including lace techniques that in the instance of “filet look alikes” introduced superimposing designs as a quicker method to achieve such fabrics. The common motifs used were often that of a heart or a rose.
One of the many confusing things in lace, is that the punched holes or mylar squares do not represent actual holes in the final fabric. Alternate rows of holes represent first transfers to left, then transfer to right. Brother and Studio punchcard sets included with purchase of machines both include pre-punched cards suitable for this type of mesh. Numbers sometimes varied with machine model year. Studio No. L-6, Brother No 17J (also 20G etc.) are 2 such samples and are vertical mirrored images of each other. Superimposed motifs constitute blank areas of card. Depending on preference some readjustments may be required after a test swatch to alter placement of some of the mesh holes. My fabric below was knit using the basic faggot lace Brother mesh, the corresponding “card” close to 180 rows in length to achieve the brick repeat.
a portion of the card
sideways view of resulting fabric, knit side facing
